Layered Manufacturing (LM) techniques build a part by adding thin layers of material. In this process, overhangs need to be supported by sacrificial supports, resulting in an increase in the build time, wastage of material, and costly post-processing. Metal-based LM machines with the capability to deposit material along multiple directions resolve most of the above problems. Importantly, these machines can deposit nonplanar slices. In this paper, we study such slices and present a task framework for their use with multidirectional layered deposition machines. The aim of the analysis is to identify part subvolumes that can be built using nonplanar slices for a process-dependent overhang angle. Solution methodologies for 2D, extruded parts, and general 3D parts are presented. Algorithms and illustrative example parts are included.